Finding Idempotents? I was wondering if there is a method for finding primitive idempotents of a finite dimensional algebra (over a field)? or in other words is there any way to build the complete set of primitive idempotents from one (idempotent)?
Or if it is impassible to find a method,
I believe a good reference or some tricks (general notes) on idempotents is also going to be very useful to me.   
 A: The only general method for finite dimensional algebras would be to look at a general element in the algebra and see what constraints it would have to fulfill to satisfy $x^2=x$. And of course, for any idempotent $e$, $1-e$ is an idempotent, so if you get bored with one you can jump to the other.
Of course, certain things will help simplify things. If you can decompose your ring into a direct product of rings, you know that the identities of these rings are idempotents which add up to the identity of the product ring.
Once you find a nontrivial idempotent $e$, you can compute what $eRe$ looks like. If $eRe$ is a local ring, then you're good because $e$ is a primitive idempotent and you can move on. If it isn't local, that means $e$ must decompose in $eRe$, and so you repeat the process of finding nontrivial idempotents in $eRe$.
Now I see you've tagged the question with representation theory, so chances are good you might be working with group algebras like $F[G]$. If so, I have a few more tricks up my sleeve for you.
One of the most useful is that for any subgroup $H<G$ ($G$ finite, and let's require the characteristic of $F$ to be zero while we're at it), the sum of the elements in $H$ is an idempotent of $F[H]$. And if I remember correctly, if $H$ is normal, then the idempotent is central. This even applies to the subgroup $G$, where I believe the resulting idempotent is the identity element of $F$ in the decomposition $F[G]\cong F\oplus \omega(G)$. 
Finally, for the special case of group rings over cyclic groups and $\Bbb C$, Thomas Andrews showed me some neat tricks using roots of $x^n-1$ for generating idempotents. This might be generalizable. I invite you to read about it in the comments here.
A: See p28 The Theory of Group Characters and Matrix Representations of Groups by D E Littlewood. But he makes it sound difficult!
